Well, I will try to give some intuition close to the one of Wikipedia as you may desire. The PR-AUC can be thought of as the probability that a classifier will rank a randomly chosen "positive" instance (from the retrieved predictions) higher than a randomly chosen "positive" one (from the original positive class). It should be noted that this is based on my own interpretation and may be subject to error.
In another Wikipedia page, the following text is relevant "precision (also called positive predictive value) is the fraction of retrieved instances that are relevant, while recall (also known as sensitivity) is the fraction of relevant instances that are retrieved".
PR-Curve is a very important metric, especially, when dealing with imbalance datasets. I would advise looking at this study for further details.
From a different persepctive, I would say that when both sensitivity and precision are of importance to the experimenter, then we can think of them as exploration and exploitation terms, respectively. Basically, you may limit strictly the threshold of predictions over the positive class, allowing a very high precision but at the cost of less exploration (i.e. lower sensitivity) (or new insights about the the positive class). Relaxing this constraint, can allow for exploring those cases that we predict as positives and never thought they are.
One would want to always maximize both but for certain cases, this might be very difficult and in some applications, sacrificing one towards enhancing the other is preferable.